The above sets of complex problems are dealt with in this study in a simplified

manner, with the following objective and associated tasks.

1.2 Objective and Tasks

The main objective of this study is to examine the hydraulics and thence the

stability of a system of two sandy ocean inlets connected to interconnected bays. The

sequence of tasks carried out to achieve this goal is as follows:

1 Deriving the basic hydraulic equations using the linearized approach for a
complex four inlets and three bays system.

2 Solving these equations, applying them to the St. Andrew Bay system, and
comparing the results with those obtained from the hydrographic surveys.

3 Developing stability criteria using the basic Escoffier (1940) model for one inlet
and one bay and then extending this model to the two inlets and a bay.

4 Carrying out stability analysis for N inlets and a bay using the linearized lumped
parameter model of van de Kreeke (1990), and then applying it to the St. Andrew
Bay system.

1.3 Thesis Outline

Chapter 2 describes the hydraulics of the multiple inlet-bay system. It progresses

from the basic theory to the development of linearized models for simple and complex

systems. Chapter 3 describes the stability of the system, including an approximate

method to examine multiple inlets in a bay. Chapter 4 includes details of hydrographic

surveys and summarizes the data. Chapter 5 discusses the input and output parameters

required for the calculation. It also presents the results. All calculations are given in the

appendices. Conclusions are made in Chapter 6, followed by a bibliography and a

biographical sketch of the author.

CHAPTER 2
HYDRAULICS OF A MULTIPLE INLET BAY SYSTEM

2.1 Governing Equations of an Inlet-Bay System

2.1.1 System Definition

The governing equations for a simple inlet-bay system may be derived by

considering the inlet connecting the ocean and the bay as shown in Figure 2.1.

SBay

11o Ocean
Figure 2.1 One bay and one inlet system

These equations are derived subjected to the following assumptions.

1 The inlet and bay banks are vertical.

2 The range of tide is small as compare to the depth of water everywhere.

3 The bay surface remains horizontal at all times, i.e., the tide is "in phase" across
the bay. That means the longest dimension of the bay be small compared to the
travel time of tide through the bay.

4 The mean water level in the bay equals that in the ocean.

5 The acceleration of mass of water in the channel is negligible.

6 There is no fresh water inflow into the bay.

7 There is no flow stratification due to salinity.

8 Ocean tides are represented by a periodical function.

2.1.2 Energy Balance

Applying the energy balance between ocean and bay one gets

2 2
77 +a, = r7 + a, +-Ah (2.1)
2g 2g

where

ro = Ocean tide elevation with respect to mean sea level,

7B = Bay tide elevation with respect to mean sea level,

Uo = Ocean current velocity,

UB = Bay current velocity,

a, and as = Coefficients greater than one which depend on the spatial distribution

of Uo and UB, respectively,

EAh = Total head loss between the ocean and the bay, and

g = acceleration due to gravity.

It is also assumed that ocean and bay are relatively deep; thus Uo and uB are small enough

to be neglected. Then Eq. (2.1) becomes

Ah = o7 -rB (2.2)

There are generally two types of head losses. One includes concentrated or

"minor losses" due to convergence and divergence of streamlines in the channel. The

second type is gradual loss due to bottom friction in the channel. The entrance and exit

2
losses may be written in terms of the velocity head in the channel, with the entrance
2g

loss coefficient ken and the exit loss coefficient kex, i.e.,

2
Entrance loss = ke (2.3)
2g

2
Exit loss = ke (2.4)
2g

where uc is the velocity through the inlet. Gradual energy losses per unit length depend

on the channel roughness and are given in form of Darcy-Weisbach friction factor

J u2
Gradual loss = (2.5)
4R 2g

where

f= Darcy-Weisbach friction coefficient,

R = hydraulic radius of channel, and

L = Length of channel.

Substitution of Eqs. (2.3), (2.4) and (2.5) into (2.2) gives

1o 7 -i k + ex + (2.6)
2g 4R

or

uc= -g 7o l .sign(o Bn) (2.7)
ke +k +
4R

The sign( ro-r/B) term must be included since the current reverses in direction every half

tidal cycle.

2.1.3 Continuity Equation

The equation of continuity, which relates the inlet flow discharge to the rate of

Figure 3.5 Possible configurations of equilibrium flow curves for a two-inlet bay system.
Stable equilibrium flow area is represented by 0 and unstable equilibrium is
represented by 0 The hatched area in (a) represents the domain of the stable
equilibrium flow area (source: van de Kreeke, 1990)

represented by the hatched rectangle in Figure 3.5 (a). The general shapes of the

equilibrium flow curves and their relative positions in the (Ai, A2) plane are presented in

Figure 3.5. The detailed explanations to the Figure 3.5 are given in Appendix D.

3.3 Stability Analysis with the Linearized Model

Due to the complex nature of sediment transport by waves and currents it is

difficult to carry out an accurate analysis of the stability of single or multiple inlet

systems. We will therefore attempt to carry out an approximate analysis based on the van

de Kreeke (1990) linearized lumped parameter model.

The justification for use of simple model is that for purpose of this study the

stability analysis serves to illustrate a concept rather than to provide exact numerical

results. Accurate numerical values can only be obtained by using a full-fledged two-

dimension tidal model to describe the hydrodynamics of the bay.

3.3.1 Linearized lumped parameter model for N Inlets in a Bay

The basic assumptions of the Linearized lumped parameter model are as follows:

1 The linearized model assumes that the ocean tide and the velocity are simple
harmonic functions.

2 The water level in the bay fluctuates uniformly and the bay surface area remains
constant.

3 Hydrostatic pressure, and shear stress distribution along the wetted perimeter of
the inlet cross-section is uniform.

4 For a given bay area and inlet characteristics, the tidal amplitude and/or tidal
frequency must be sufficiently large for equilibrium to exist. Similarly, larger the
littoral drift due to waves, larger the equilibrium shear stress required to balance it
and therefore the equilibrium velocity, the larger the required bay surface area,
tidal amplitude and the tidal frequency or, in other words, Eq. (3.17) and Eq.
(3.19) must be satisfied for the existence of equilibrium areas.

5 There is no fresh water discharge in the bays.

6 In a shallow bay the effect of dissipation of tidal energy cannot be ignored,
especially if the bay is large.

Inlet flow dynamics of the flow in the inlets are governed by the longitudinal

pressure gradient and the bottom shear stress, van de Kreeke (1967),

0 =- (3.2)
p ix ph

in which is the pressure, p is the water density, h is the depth and r is the bottom shear

stress. This stress is related to the depth mean velocity u

r = pFu u (3.3)

where F=f/8, is the friction coefficient. Integration of Eq. (3.2) (with respect to the

longitudinal x-coordinate) between the ocean and the bay yields (van de Kreeke 1988).

u I U -= 2gR (o ) (3.4)
mjRm + 2F1L

In Eq. (3.4), u, refers to the cross-sectional mean velocity of the ith inlet, g is the

acceleration due to gravity, m, is the sum of exit and entrance losses, R, is the hydraulic

radius of the inlet, L, is the length of the inlet, ro is the ocean tide, and 7B is the bay tide.

The velocity u, is positive when going from ocean to bay.

Assuming the bay surface area to fluctuate uniformly, flow continuity can be

expressed as

u, A,= d4B (3.5)
=1 dt

in which A, is the cross-sectional area, AB is the bay surface area and t is time.

Considering u, to be a simple harmonic function of t, Eq. (3.4) is linearized as

shown in Appendix B to yield

8 2gR,
Umaxu,( = (7 ) (3.6)
37r mjR + 2FL

in which uax, is the amplitude of the current velocity in the ith inlet. It follows from Eq.

in which sets of A, are given by Eq. (3.18) (as we have two real and positive solution for
A,). When any A, is known, the cross-sectional areas of the other (N-l) inlets follow from
Eq. (3.14) with B, given by Eq. (3.15). One root of Eq. (3.18) is always negative. The
other two are real and positive roots provided that
Aa3 3 2 5 FlL2 5 FNLN 21
A 3 > u15eq +....N (3.19)
2 37r eqa ag g j

The above stability concept, when applied to a multiple-bay inlet system,

becomes complicated because the loci of the set of the values [A1, A2....AN] for which the

tidal maximum of the bottom shear stress equals the equilibrium stress, are rather

complicated surfaces and make it difficult to determine whether inlets are in a scouring

mode or shoaling mode. With some simplifying assumptions, the stability analysis for a

multiple-inlet system can be reduced to that for a two-inlet system. This is considered

next in the context of the St. Andrew Bay system.

3.4 Application to St. Andrew Bay System

In the above model if N=2, the model can be applied to the two inlet system. The

curve. A line can be drawn passing from the intersection of two equilibrium flow areas.

Above the line Bi>B2 and therefore u
1 When the point defined by the actual cross-sectional areas [Ai, A2] is located in
the vertically hatched zone or anywhere outside the curves, (Zone-1), both inlets
close.

2 When the point is located in the crosshatched zone, (Zone-2), Inlet 1 will remain
open and Inlet 2 will close.

3 When the point is located in the diagonally hatched zone, (Zone-3), Inlet 1 will
close and Inlet 2 will remain open.

4 Finally, when the point is located in the blank zone, (Zone-4), one inlet will close
and the other will remain open. However, in this case which one closes depends
on the relative rates of scouring and or/shoaling.

The St. Andrew Bay system is similar to the case of two inlets in a bay. In reality

there are three interconnected bays, but only one is connected with the Gulf. So there is

no forcing due to ocean tide from the other two bays. Thus, all the bays collectively

behave as if there is only one bay connected by two inlets. So the linear model for N

inlets can be applied to the St. Andrew system, where N = 2. The development of

Figure 4.3 East Pass channel before it's opening in December 2001. Plan view (pre-
construction) design geometry and then anticipated current measurement
transects are shown. The dots show the new cross-section (source: Jain et al.,
2002)

4.2 Summary of Field Data

Three hydrographic surveys were done by the University of Florida's Department

of Civil and Coastal Engineering in the years 2001 and 2002. Figure 4.4 shows the

bathymetry of St. Andrew Bay Entrance and the different cross-sections measured during

the surveys. Cross-sections A-i, A-2 and B-l, B-2 were measured in September 2001,

A'-1, A'-2, B'-1, B'-2, C'-1, C'-2, in December 2001, and D-l, D-2 in March 2002. Flow

discharges, vertical velocity profiles and tide were also recorded. The tide gage (in the

September 2001 survey only) was located in waters (Grand Lagoon) close to the entrance

channel. The discharge and velocity data was measured with a vessel-mounted Acoustic

Table 5.4 provides the input values for all the three cases of the model as

described in Section 5.2.

1 The amplitude in each bay is found by applying a weighting factor proportional to
the tide station contribution to the total bay area.

2 Initial values are assumed for (ro -ri1) max, (iB1 -7B2) max (1B1 -/7B3) maxfor the
initial calculation. The September 2001 tide showed a semidiurnal signal, with a
period of 12.42 h. The tide in March 2002 showed diurnal signature with a period
of 25.82 h. The model was run three times for three different cases as described in
Section 5.2. Details regarding all input parameters are found in Jain and Mehta
(2002), and are also summarized in Chapter 4. Table 5.4 gives values of all input
parameters required for the model.

Table 5.8 Stability observations for St. Andrew Bay Entrance and East Pass.
Figure Placement of cross-sectional Observations
area pair [A1, A2], (black dot)
Figure 5.1 Zone-1 Both inlets are unstable
Figure 5.2 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.3 Zone-4 Only one is stable i.e. St. Andrewa
Figure 5.4 Zone-1 Both inlets are unstable
Figure 5.5 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.6 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.7 Zone-1 Both inlets are unstable
Figure 5.8 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.9 Zone-4 Only one is stable i.e. St. Andrewa
Figure 5.10 Zone-1 Both inlets are unstable
Figure 5.11 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.12 Zone-4 Only one is stable i.e. St. Andrewa
a As per Figure 3.6, it is clear that even in Zone-4 only one inlet is stable, this is further
clarified from Figure 3.5, which shows that only one inlet can be stable at one time.

CHAPTER 6
CONCLUSIONS

6.1 Summary

St. Andrew Bay, which is a composite of three interconnected bays (St. Andrew

Bay proper, West Bay + North Bay and East Bay) is located in Bay County on the Gulf

of Mexico coast of Florida's panhandle. It is part of a three-bay and two-inlet complex.

One of these inlets is St. Andrew Bay Entrance and the other is East Pass, which are both

connected to St. Andrew Bay on one side and the Gulf on the other. Prior to 1934, East

Pass was the natural connection between St. Andrew Bay and the Gulf. In 1934, St.

Andrew Bay Entrance (Figure 4.2) was constructed 11 km west of East Pass through the

barrier island to provide a direct access between the Gulf and Panama City. The interior

shoreline of the entrance has continually eroded since it's opening. East Pass was closed

in 1998, which is believed to be due to the opening of the St. Andrew Bay Entrance.

In December 2001, a new East Pass was opened (Figure 4.3), and the effect of this

new inlet is presently being monitored over the entire system. Accordingly, the objective

of the present work was to examine the hydraulics of the newly formed two-("ocean")

inlet/three-bay system and its hydraulic stability, especially as it relates to East Pass.

The first aspect of the tasks performed to meet this objective was the development

of equations for the linearized hydraulic model for the system of three bays and four

inlets (two ocean and two between bays), and solving and applying them to the St.

Andrew Bay system. The second aspect was the development of the ocean inlet stability

criteria using the Escoffier (1940) model for one inlet and one bay and extending this

model to the two ocean inlets and a bay. Stability analysis for the St. Andrew Bay system

was then carried out using the linearized lumped parameter model of van de Kreeke

(1990).

6.2 Conclusions

The following are the main conclusions of this study:

1 If the system is modeled as a three-bay system as compare to a one-bay system,
the error in the phase difference, SB1, decreases from 6% to 0% and in the velocity
amplitude from 3% to 2%. Moreover the error in maximum head difference, (/o -
tB1i) m, also decreases from 6% to 0%.

2 The amplitudes of velocities and bay tides are within 5%, which is a reasonably
small error band. The percent error for St. Andrew Bay is almost 0%, and for the
other bays it is within 20%.

3 The bay area has a significant effect on the stability of the two inlets. At a bay
area of 74 km2 both inlets are unstable. Increasing it by 22% to 90 km2 stabilizes
St. Andrew Bay Entrance, and by 42% to 105 km2 stabilizes East Pass as well.

4 Two inlets can never be simultaneously unconditionally stable.

5 Keeping the bay area at 105 km2 and increasing the length of East Pass from 500
m to 2000 m destabilizes this inlet because as the length increases the dissipation
in the channel increases as well.

6 A triangular channel cross-section is a better approximation than a rectangular
one, because given the same values of all other hydraulic parameters, St. Andrew
Bay Entrance with a rectangular cross-section is found to be barely stable,
whereas with a triangular cross-section it is found to be stable, as is the case.

6.3 Recommendations for Further Work

Accurate numerical values required for the stability analysis of a complex inlet-

bay system can only be obtained by using a two- (or three)-dimensional tidal model to

describe the hydrodynamics of the bay.

Freshwater discharges from the rivers into the bay should be incorporated through

numerical modeling.

75

Including a more realistic assumption for the channel cross-section can improve

the stability analysis.

APPENDIX A
ALGORITHMS FOR MULTIPLE INLET-BAY HYDRAULICS

A.1 Introduction

The linearized approach described in Chapter 2 has been used to evaluate the